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Properties

Label 2-637-13.4-c1-0-20
Degree $2$
Conductor $637$
Sign $-0.297 + 0.954i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.08 − 1.20i)2-s + (0.888 − 1.53i)3-s + (1.88 + 3.26i)4-s − 0.706i·5-s + (−3.69 + 2.13i)6-s − 4.25i·8-s + (−0.0791 − 0.137i)9-s + (−0.848 + 1.46i)10-s + (4.66 + 2.69i)11-s + 6.69·12-s + (−0.746 − 3.52i)13-s + (−1.08 − 0.627i)15-s + (−1.33 + 2.31i)16-s + (2.12 + 3.68i)17-s + 0.380i·18-s + (−2.40 + 1.38i)19-s + ⋯
L(s)  = 1  + (−1.47 − 0.849i)2-s + (0.513 − 0.888i)3-s + (0.942 + 1.63i)4-s − 0.316i·5-s + (−1.50 + 0.871i)6-s − 1.50i·8-s + (−0.0263 − 0.0456i)9-s + (−0.268 + 0.464i)10-s + (1.40 + 0.812i)11-s + 1.93·12-s + (−0.207 − 0.978i)13-s + (−0.280 − 0.162i)15-s + (−0.333 + 0.578i)16-s + (0.515 + 0.892i)17-s + 0.0895i·18-s + (−0.551 + 0.318i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.297 + 0.954i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.564483 - 0.767162i\)
\(L(\frac12)\) \(\approx\) \(0.564483 - 0.767162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (0.746 + 3.52i)T \)
good2 \( 1 + (2.08 + 1.20i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.888 + 1.53i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.706iT - 5T^{2} \)
11 \( 1 + (-4.66 - 2.69i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.12 - 3.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.40 - 1.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.35 + 5.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.27 - 2.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.74iT - 31T^{2} \)
37 \( 1 + (1.18 + 0.682i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.91 - 2.25i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.78 - 6.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.0870iT - 47T^{2} \)
53 \( 1 - 7.04T + 53T^{2} \)
59 \( 1 + (5.41 - 3.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.40 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.09 + 5.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.03 - 5.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 3.20iT - 83T^{2} \)
89 \( 1 + (1.75 + 1.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.62 + 2.67i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.28904750858393271276262850158, −9.330455572115103340047930847090, −8.675340709006182613403713226780, −7.912834274651699003109390059536, −7.27109136993487642828095951006, −6.23104727523305935577453699656, −4.44013047932770655764376037421, −3.00403115669968970027781138503, −1.91760005866861310135576992368, −1.01449087543347307547984978665, 1.24013701841236827611023724002, 3.14089306609012690805132278459, 4.29435250442663339830815351009, 5.71141904273754023697597489188, 6.84626633368087652248092690746, 7.21079559315608976332784770041, 8.772813699039879029496098135850, 8.919541557868467702311830542705, 9.599064715486696600491222926251, 10.46049813374540912166531784363

Graph of the $Z$-function along the critical line